Uniform Convergence of Martingales

Abstract

The uniform law of large numbers, or equivalently the law of large numbers in Banach spaces, has been studied intensively in the past two decades, e.g. see [2] and [4]. Suppose that ξ₁, ξ₂,… are independent identically distributed random variables taking values in a measurable space (S, S), and let f: S × T→ ℝ be a given function, where T is a given set. Suppose that m(t) = Ef(ξ₁, t) exists for all t ∈ T, then in [2] and [4] you may find a series of necessary and sufficient conditions for the following form of the uniform law of large numbers:

$${1 \over n}\sum\limits_{j = 1}^n {f\left( {{\xi _j},t} \right) \to m\left( t \right)} \,\,\,\,\,{\rm{uniformly}}\,{\rm{on}}\,T\,{\rm{a}}{\rm{.s}}{\rm{.}}$$